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f0f05de | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | import OAI.Combinatorics.BalancedRyser.Local.LocalPrescriptionCounting
namespace OAI
namespace Balanced.Local
noncomputable section
attribute [local instance] Classical.propDecidable
open scoped BigOperators
variable {F : Type*} [Field F] [Fintype F] [DecidableEq F]
variable {b κ : ℕ}
theorem alignment_and_badCover_probability (P : EqualityPattern b κ)
(a : DirectionAssignment F b κ) (bits : Bits b) (p : Point F) :
offsetProbability (fun M => Aligned P a bits p M ∧ AdmissibleBadCover P a bits p M) =
offsetProbability (AdmissibleBadCover P a bits p) := by
congr 1
funext M
apply propext
exact ⟨And.right, fun h => ⟨h.1.1, h⟩⟩
theorem badCover_probability_le_of_conditioned (P : EqualityPattern b κ)
(a : DirectionAssignment F b κ) (bits : Bits b) (p : Point F) (t : ℝ)
(hnorm : offsetProbability (Aligned P a bits p) =
1 / (Fintype.card F : ℝ) ^ (4 * b))
(hcond : conditionedOffsetProbability (Aligned P a bits p)
(AdmissibleBadCover P a bits p) ≤ t) :
offsetProbability (AdmissibleBadCover P a bits p) ≤
1 / (Fintype.card F : ℝ) ^ (4 * b) * t := by
unfold conditionedOffsetProbability at hcond
rw [alignment_and_badCover_probability, hnorm] at hcond
have hp : (0 : ℝ) < 1 / (Fintype.card F : ℝ) ^ (4 * b) := by positivity
simpa only [mul_comm] using (div_le_iff₀ hp).mp hcond
omit [Field F] [DecidableEq F] in
theorem card_directionAssignment (hk : κ ≤ 4 * b) :
Fintype.card (DirectionAssignment F b κ) =
(Fintype.card F + 1) ^ (5 * b - κ) := by
simp only [DirectionAssignment, Fintype.card_prod, Fintype.card_fun,
Fintype.card_fin, Direction, Fintype.card_option]
rw [← pow_add]
congr 1
omega
theorem directionBadCoverMass_le_of_conditioned
(P : EqualityPattern b κ) (bits : Bits b) (p : Point F)
(hk : κ ≤ 4 * b) (t : ℝ) (ht : 0 ≤ t)
(hnorm : ∀ a : DirectionAssignment F b κ, Eligible P a →
offsetProbability (Aligned P a bits p) =
1 / (Fintype.card F : ℝ) ^ (4 * b))
(hcond : ∀ a : DirectionAssignment F b κ, Eligible P a →
conditionedOffsetProbability (Aligned P a bits p)
(AdmissibleBadCover P a bits p) ≤ t) :
directionBadCoverMass P bits p ≤
((Fintype.card F + 1 : ℕ) : ℝ) ^ (5 * b - κ) /
(Fintype.card F : ℝ) ^ (4 * b) * t := by
unfold directionBadCoverMass
simp only [alignment_and_badCover_probability]
calc
_ ≤ ∑ _a : DirectionAssignment F b κ,
1 / (Fintype.card F : ℝ) ^ (4 * b) * t := by
apply Finset.sum_le_sum
intro a _
by_cases he : Eligible P a
· rw [ite_eq_left he]
exact badCover_probability_le_of_conditioned P a bits p t (hnorm a he) (hcond a he)
· rw [ite_eq_right he]
positivity
_ = _ := by
rw [Finset.sum_const, Finset.card_univ, nsmul_eq_mul, card_directionAssignment hk]
push_cast
ring
theorem bad_pencil_probability_le_of_conditioned
(Q : RetentionRule F) (w : Kernel F) (K : ℝ) (hK0 : 0 ≤ K)
(hK : KernelConditions Q w K) (p : Point F) (hb : 2 ≤ b) (hk : κ ≤ 2)
(t : ℝ) (ht : 0 ≤ t)
(hnorm : ∀ (P : EqualityPattern b κ) (a : DirectionAssignment F b κ), Eligible P a →
∀ bits : Bits b, offsetProbability (Aligned P a bits p) =
1 / (Fintype.card F : ℝ) ^ (4 * b))
(hcond : ∀ (P : EqualityPattern b κ) (a : DirectionAssignment F b κ), Eligible P a →
∀ bits : Bits b, conditionedOffsetProbability (Aligned P a bits p)
(AdmissibleBadCover P a bits p) ≤ t) :
selectedProbability w (fun C => C.splitMultiplicity p = b ∧
reuseCount C p = κ ∧ BadLocalCoverAt b p C) ≤
(4 * (b : ℝ)) ^ 4 * (1024 * K) ^ b / (b.factorial : ℝ) *
(t / (Fintype.card F : ℝ) ^ κ) := by
apply bad_pencil_probability_le_common_factor Q w K hK0 hK p hb hk t ht
intro P bits
exact directionBadCoverMass_le_of_conditioned P bits p (by omega) t ht
(fun a ha => hnorm P a ha bits) (fun a ha => hcond P a ha bits)
end
end Balanced.Local
end OAI
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